Most lists of things to do in Tbilisi start with the Old Town, and for once the internet is right. The difference is how you do it. Tourists walk up to Narikala Fortress in the midday heat, arrive sweaty, and leave in ten minutes. Locals take the cable car up from Rike Park and walk down. Do it that way.
The cable car costs 2.5 GEL (about $1) each way, and you'll need a Metromoney transport card from the ticket window (2 GEL, reusable on the metro and buses). The ride takes two minutes and crosses the Mtkvari River with the whole old city of Tbilisi spread out below you. At the top you get Narikala, a fortress that has stood on this ridge in some form since the 4th century. Entry is free. The walls are partly ruined and there are no barriers in places, so watch your step with kids.
Walk five minutes along the ridge to Kartlis Deda, the 20-meter aluminum Mother of Georgia statue. She holds a bowl of wine for guests and a sword for enemies, which is honestly a fair summary of the national character. Then take the path down through the Betlemi district. This is the part most visitors miss. Crooked wooden balconies, cats on every second doorstep, tiny courtyards where laundry hangs between 200-year-old houses. It looks staged. It isn't, people live here.
The walk down deposits you near Abanotubani and the Leghvtakhevi gorge, where a small waterfall hides about 300 meters behind the bath houses. Yes, a waterfall in the middle of a capital city. The boardwalk to it is free and takes ten minutes.

